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International Journal Of Engineering And Computer Science ISSN: 2319-7242
Volume 4 Issue 12 Dec 2015, Page No. 15123-15131
Performance Measures for a three-unit compact circuit
Ashish Namdeo1 ,V.K. Pathak2 & Anil Tiwari3
Comp-Tech Degree College, Sorid Nagar, Dhamtari(Chhattisgarh), India
B.C.S. Govt. P.G. College, Dhamtari PIN 493773, (Chhattisgarh), India
Disha College, Ramnagar Kota, Raipur, (Chhattisgarh), India
Abstract
This paper analyses a three component system with single repair facility. Denoting the failure times
of the components as T1 and T2 and the repair time as R, the joint survival function of (T1, T2, R) is assumed
to be that of trivariate distribution of Marshall and Olkin. Here, R is an exponential variable with parameter
α and T1 and T2 are independent of each other. In this paper use of Laplace-Transform is taken for finding
Mean Time Between Failure, Availability and Mean Down Time and table for Reliability measure is shown
in the end.
Key Words : MTBF, Availability, MDT, Reliability.
[1] Introduction
Reliability measures for a two-component standby system with repair facility were obtained
by several authors under different assumptions in the past. Lie et al. (1977) and Yearout et al. (1986) have
done extensive reviews for the failure times and repair times assuming that these are statistically
independent. Joshi and Dharmadhikari (1989) considered the bivariate exponential distribution to derive the
performance measures associated with a two-component standby system. Goel and Srivastava (1991)
considered a correlated structure for the failure and repair times and obtained various reliability measures.
In many situations, a unit or system can be repaired immediately after breakdown. In such
cases, the mean time between failures refers to the average time of breakdown until the device is beyond
repair. When a system is often unavailable due to breakdowns and is put back into operation after each
breakdown with proper repairs, the mean time between breakdowns is often defines as the mean time
between failures. If we consider only active repair time i.e. the time spent for actual repair, then the mean
time to repair (MTTR) is the statistical mean time for active repair. It is the total active repair time during a
given period divided by the number of during the same interval. Frequently, a system may be unavailable on
account of periodic inspections and not because of breakdowns. By the systematic inspection or preventive
maintenance for the detection of defects and prevention of failures, the system is kept in a satisfactory
Ashish Namdeo1 IJECS Volume 04 Issue 12 December 2015, Page No.15123-15131
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operational condition. The time spent for this is termed as the preventive maintenance downtime. There is
difference between mean time between maintenance (MTBM) and mean time between failures (MTBF).
When preventive maintenance downtime is zero or is not considered, MTBM is same as MTBF.
Here in this paper, we perform the analysis of a three-component standby system with single
repair facility. Here T1 is the exponential variable with parameter k1 and T2 is another random variable with
parameter k2. R is an exponential variable with parameter α . T1 and T2 are independent of each other. It is
further assumed that these components are identical in nature and each unit works as new after the repair and
switching devices are perfect and instantaneous.
The following are the assumptions for the model:
(i)
The system is composed of three components linked in parallel-series configuration (Fig. 1).
(ii)
The components are non-identical in nature.
(iii)
At time t=0, all the components are in operable mode.
(iv)
After repair each unit works as new.
(v)
Switching devices are perfect and instantaneous.
Define pi (t )  Pr{X (t )  i : X (0)  0}, i  0 , 1, 2
Here in this model, we consider the following trivariate exponential distribution for (T1, T2, R) with survival
function of (Marshall and Olkin, 1967) of the form :
F (t1 , t 2 , t 3 )  exp[ k1 (t1  t 2 )  k 2 t 3  k 3 {max( t1 , t 2 )  max( t 2  t 3 )}]
t1 , t 2 , t 3  0 ; k1 , k 2  0 ; k 3  0
[1.1]
It may be observed that
i)
T1 and T2 are independent and identically distributed exponential random variables with the
parameter (k1+k2),
ii)
R is exponential with the parameter (k2+2k3), not necessarily independent of (T1,T2) and
iii)
(T1,R) and (T2,R) are identically distributed as bivariate exponential with the parameter (k1,
k2+k3, k3).
By considering (1) as the survival function of (T1,T2,R) , we obtain expressions for reliability,
MTBF and the gain due to repair facility. The state transition diagram for the system, in the
interval (t , t  t ) is given below:
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Unit II
Unit I
Unit III
Figure 1 State Transition Diagram of three-unit standby System
[2] MTBF Calculation
Since reliability of the system is given by R(t)= P0(t) + P1(t) + P2(t), we want to find the expressions for

P0(t), P1(t) and P2(t). We know that MTBF =
 R(t ) dt
0
We have, pi (t )  pr {X (t )  i : X (0)  0} , i  0,1, 2
So, pi (t   t )  pr {X (t   t )  i : X (0)  0}
2
  p r { X (t   t )  0 , X (t )  j : X (0)  0}
j 0
Using P( A B )  P(A / B) P( B) , we have
2
  p r { X (t   t )  0 , X (t )  j}{X (t )  j : X (0)  0}
j 0
2
  p r { X (t   t )  0 , X (t )  j} p j (t )
j 0
 pr { X (t   t )  0 : X (t )  0} p0 (t )  pr { X (t   t )  0 : X (t )  1} p1 (t )
 pr { X (t   t )  0 : X (t )  2} p2 (t )
p0 (t   t )  (1    t ) p0 (t )    t p1 (t )  0( t )
p0 (t   t )  p0 (t )    t p0 (t )    t p1 (t )  0( t )
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Dividing by  t and taking limit t  0 we have
Lim
t  0
p0 (t   t )  p0 (t )
0( t )
  p0 (t )   p1 (t )  Lim
 t0  t
t
p0/ (t )   p0 (t )   p1 (t )
[2.1]
Similarly, we can write p1 (t   t )  pr ( X (t   t )  1: X (0)  0}
2
p1 (t   t )   pr ( X (t   t )  1 , X (t )  j : X (0)  0}
j 0
2
  pr ( X (t   t )  1 , X (t )  j } p j (t )
j 0
 pr{ X (t   t )  1: X (t )  0} p0 (t )  pr{ X (t   t )  1: X (t )  1} p1 (t )
 pr{ X (t   t )  1: X (t )  2} p 2 (t )
so, we have p1 (t   t )    t p0 (t )  {1  (   )  t} p1 (t ) which on simplification gives
p1/ (t )   p0 (t )  (   ) p1 (t )
Assuming,   k1  k 3 ,   k1 and   k 2  k 3 , we get the equations [6.2.3] and [6.2.4] in the following
reduced form:
p0/ (t )  (k1  k3 ) p0 (t )  (k 2  k3 ) p1 (t )
p1/ (t )  (k1  k3 ) p0 (t )  (k1  k 2  k3 ) p1 (t )
p2/ (t )  k 2 p0 (t )  (k1  k 2  k3 ) p1 (t )  k 2 p2 (t )
[2.2-2.4]
Taking Laplace transform on both the sides of [6.1.5] and [6.1.6] and noting that L{ pi (t )}  Li (s)
We get (k1  k 3  s) L0 (s)  (k 2  k 3 ) L1 (s)  (k 3  s) L2 (s)  1
(k1  k3 ) L0 (s)  (k1  k 2  k3  s) L1 (s)  k 2 L2 (s)  0
and (k 2  s) L0 (s)  (k1  s) L1 (s)  (k1  k3  s) L2 (s)  0
solving using Cramer rule, we get
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DOI: 10.18535/Ijecs/v4i12.6
L0 ( s ) 
L2 ( s ) 
( k1  k 2  k 3  s )
{s  (2k1  k 2  2k 3 ) s  k1 (k1  k 3 )}
2
L1 ( s) 
(k1  k 3 )
and
{s  (2k1  k 2  2k 3 ) s  k1 (k1  k 3 )}
2
(k 2  k 3  s )
{s  (2k1  k 2  2k 3 ) s  k1 (k1  k 3 )}
2
Let s1 and s2 be the roots of the equation s 2  (2k1  k 2  2k 3 )s  k1 (k1  k 3 )  0
let
s1 
and s 2 
 (2k1  k 2  2k 3 )  k 22  4(k1  k 3 )(k 2  k 3 )
2
[2.5-2.6]
 (2k1  k 2  2k 3 )  k 22  4(k1  k 3 )(k 2  k 3 )
2
Since s1 , s2 <0 . Thus we have,
L0 ( s) 
( k1  k 2  k 3  s )
;
{(s  s1 )( s  s 2 )}
L1 ( s) 
(k1  k 3 )
{(s  s1 )( s  s 2 )}
;
L2 ( s) 
(k 2  k 3  s)
{(s  s1 )( s  s 2 )}
Resolving into partial fractions, we have
L0 ( s) 
(k1  k 2  k 3  s1 ) (k1  k 2  k 3  s 2 )

( s  s1 )(s1  s 2 )
( s  s 2 )(s1  s 2 )
L1 ( s) 
(k1  k 3 )
{(s  s1 )( s  s 2 )}
L2 ( s) 
(k 2  k 3 )
{(s  s1 )( s  s 2 )}
[2.7-2.9]
Taking inverse Laplace transforms of the above equations , we get
p0 (t ) 
(k1  k 2  k 3  s1 )e s1t  (k1  k 2  k 3  s 2 )e
s1  s 2
s2t
and
(k1  k 3  s1 )(e s1t  e s2t )
p1 (t ) 
s1  s 2
p 2 (t ) 
(k 2  k 3 )(e s1t  e s2t )
s1  s 2
[2.10-2.12]
s 2 e s1t  s1e s2t
Hence the reliability of the system is given by R(t )  p0 (t )  p1 (t )  p 2 (t ) 
s1  s 2
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DOI: 10.18535/Ijecs/v4i12.6

MTBF =
 R(t ) dt  
0
So, MTBF =
( s1  s 2 )
where s1 and s2 are given by equation above.
s1 s 2
(k1  k 2  2k3 )
k1 (k1  k3 )
[2.11]
it may be noted that MTBF when there is no repair facility is given by
MTBF (no repair facility) = E (T1+T2) = E (T1) + E (T2) =
1
1
2


k1  k 3 k1  k 3 k1  k 3
[2.12]
[3] Availability analysis of the system
In this section, we consider the transient solution of the system and the availability measures
such as the point wise availability and the steady-state availability by considering the above model. By
considering the equation [1.1] as the survival function of (T1, T2, R), we obtain the expressions for point
wise availability and the steady-state availability. Using similar arguments as in the case of MTBF, we
obtain the following differential equations:
p0/ (t )  (k1  k3 ) p0 (t )  (k 2  k3 ) p1 (t )
p1/ (t )  (k1  k3 ) p0 (t )  (k1  k 2  k3 ) p1 (t )  (k 2  2k3 ) p2 (t )
p2/ (t )  k1 p1 (t )  (k 2  2k3 ) p2 (t )
[3.1-3.3]
Taking Laplace transforms of above three equations and applying the Cramer’s rule, we get
s 2  (k1  2k 2  2k 3 ) s  (k 2  k 3 )(k 2  2k 3 )
L0 ( s)  3
s  2(k1  k 2  2k 3 ) s 2  {(k1  k 2  2k 3 ) 2  k1 (k 2  k 3 )}s
L1 ( s) 
(k1  k 3 )(k 2  2k 3  s)
s  2(k1  k 2  2k 3 ) s 2  {(k1  k 2  2k 3 ) 2  k1 (k 2  k 3 )}s
L2 ( s) 
k1 (k1  k 2 )
s  2(k1  k 2  2k 3 ) s  {(k1  k 2  2k 3 ) 2  k1 (k 2  k 3 )}s
3
3
and
2
[3.4-3.6]
Let s1 and s2 be the roots of the equation
s 2  2(k1  k 2  2k3 )s  (k1  k 2  2k3 ) 2  k1 (k 2  k3 )  0
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Resolving into partial fractions, we have
s 2 {s12  (k1  2k 2  2k 3 ) s1  (k 2  k 3 )(k 2  2k 3 )} s1{s 22  (k1  2k 2  2k 3 ) s 2  (k 2  k 3 )(k 2  2k 3 )}

( s1  s 2 )(s  s1 ) s1 s 2
( s 2  s1 )(s  s 2 ) s1 s 2
L0 (s) 

(k 2  k 3 )(k 2  2k 3 )
s s1 s 2
Similarly, we can
write
L1 ( s) 
(k1  k 3 )[s 2 (k 2  2k 3  s1 )]
s ( k  2k 3  s 2 )
( k  2k 3 )
 1 2
 2
( s1  s 2 )(s  s1 ) s1 s 2
( s 2  s1 )(s  s 2 ) s1 s 2
s s1 s 2
L2 ( s) 
s 2 k1 (k1  k 3 )
s1
1


( s1  s 2 )(s  s1 ) s1 s 2 ( s 2  s1 )(s  s 2 ) s1 s 2 s s1 s 2
[3.7- 3.8]
Taking Inverse Laplace transform on both the sides of equations [6.2.7-6.2.8], we get
p0 (t ) 
s 2 e s1t {s1 ( s1  k1  2k 2  3k 3 )  (k 2  k 3 )(k 2  2k 3 )}
s1 s 2 ( s1  s 2 )
s1e s2t {s 2 ( s 2  k1  2k 2  3k 3 )(k 2  k 3 )(k 2  2k 3 )} (k 2  k 3 )(k 2  2k 3 )


s1 s 2 ( s 2  s1 )
s1 s 2
s 2 e s1t ( s1  k 2  2k 3 )(k1  k 3 ) s1e s2t ( s 2  k 2  2k 3 )(k1  k 3 ) (k 2  2k 3 )
p1 (t ) 


s1 s 2 ( s1  s 2 )
s1 s 2 ( s 2  s1 )
s1 s 2
p 2 (t ) 
s 2 e s1t k1 (k1  k 3 ) s1e s2t k1 (k1  k 3 ) k1 (k1  k 3 )


s1 s 2 ( s1  s 2 )
s1 s 2 ( s 2  s1 )
s1 s 2
The point availability of the system is given as
A(t )  1 
[3.9-3.11]
A (t) =p0(t)+p1(t)= 1-p2(t) i.e.
s 2 e s1t k1 (k1  k 3 ) s1e s2t k1 (k1  k 3 ) k1 (k1  k 3 )


s1 s 2 ( s1  s 2 )
s1 s 2 ( s 2  s1 )
s1 s 2
[3.12]
Thus the steady state availability of the system is given as
A  Lim A(t )  1 
t 
k1 (k1  k 3 )
(k 2  2k 3 )(k1  k 2  2k 3 )

s1 s 2
(k 2  2k 3 ) 2  k1 (k1  k 2  3k 3 )
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[3.13]
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DOI: 10.18535/Ijecs/v4i12.6
[4] Mean Down Time Calculation: The system mean down time is an important aspect of Availability
1  A
) . So, using the results from [3.10] and [3.13],
analysis and is evaluated by the formula MDT  MTBF (
A
we get
MDT 
2k1  k 2  2k 3
(k1  2k 3 )(2k1  k 2  2k 3 )
[4.1]
[5] References:
1. Nakagawa, T. and Osaki, S.[1976]. “Markov renewal process with some non-regenerative points and
their applications to reliability”. Microelectronics Reliability, 15(6), 633-636.
2. C.H. Lie, C.L. Hwang and F.A. Tillman [1977],“Availability of Maintained systems: A state of the
art survey”, A.I.I.E. Trans.,9, 247-259.
3. Okumoto, K. [1985]. “Availability of 2-component dependent system”. IEEE Trans. On Reliability,
R-30(2) 205-210.
4.
Yearout , R.D, Reddy P, Grosh DL [1986]. “Standby redundancy in reliability- a review”. IEEE
Trans on Reliability.
5. Joshi, B.H. and Dharmadhikari, A.D. [1989]. “Maximization of availability of 1-out-of-2 : G
repairable dependent system”. Adv. Appl. Prob. 21, 717-720.
6. Goel, L.R. and Srivastava, P. [1991]. “Profit analysis of a two-unit redundant system with provision
for rest and correlated failure and repair”. M. & R. 31, 827-833.
7. Papageorgiou, Effii and Kokolakis, George[2004]. “A Two unit parallel system supported by (n-20
standbys with general and non-identical life times”. International Journal of System Science,35(1)
(2004), 1-12.
8. Singh,S.N. & Singh,S.K.[2007]. “A study on a two unit parallel system with Erlangian repair time”,
Statistica (Bologona),67 (1) (2007), 69-77.
9. Wu,Qingtai, Zhang,Jin and Tang,Jiashan.[2015]. “Reliability evaluation for a class of multi unit cold
standby systems under Poisson shocks”. Comm.Statist.Theory Methods , 44(15) , 3278-3288.
[6] Observation:
We provide the data in the following table. The table gives the values for MDT for various values for k1, k2
and k3.
Table 6.1
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DOI: 10.18535/Ijecs/v4i12.6
k1
k2
k3
S1
S2
R(t)
.01
0.1
.01
-5.984
-6.293
12.992e-2t-10.526e-2.5t-.02e-7.876t
-3.123
-5.603
10.965e-4t-0.785e-6.5t-.802e-4.976t
-2.136
-1.610
9.095e-6t-0.584e-7.5t-.04e-9.872t
-7.000
-6.177
3.09e-2.07t-5.506e-2.5t-.076e-6.874t
-6.667
-5.986
2.654e-4.24t-1.085e-6.86t-2.202e-0.906t
-3.125
-2.955
4.889e-6.76t-0.004e-2.56t-3.045e-6.952t
-6.875
-6.172
4.225e-2.0t-3.793e-7.5t-.068e-6.07t
-4.414
-4.133
6.259e-7.25t-3.005e-2.85t-2.762e-0.006t
-3.250
-3.074
7.809e-3.75t-0.974e-2.06t-0.049e-0.952t
-7.224
-6.179
10.09e-5.07t-5.506e-2.5t-.084e-0.874t
-4.454
-3.264
12.690e-9.24t-7.025e-9.82t-9.002e-0.006t
-1.667
-1.633
14.249e-0.76t-0.904e-8.56t-3.045e-6.952t
.02
.03
.04
0.2
0.3
0.4
.02
.03
.04
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